Essays in Asset Pricing

Transcription

1 Essays in Asse Pricing Sanislav Khrapov A disseraion submied o he faculy of he Universiy of Norh Carolina a Chapel Hill in parial fulfillmen of he requiremens for he degree of Docor of Philosophy in he Deparmen of Economics. Chapel Hill 011 Approved by: Eric Renaul, Advisor Eric Ghysels Jonahan Hill Saraswaa Chaudhuri Riccardo Colacio

2 Absrac STANISLAV KHRAPOV: Essays in Asse Pricing. Under he direcion of Eric Renaul. In chaper 1 I consider a discree-sae economy and consruc an asse pricing model for he valuaion of consumpion and dividend cash flows wih shor and long mauriies. I consider hree uiliy funcions: expeced uiliy, Epsein-Zin EZ, and generalized disappoinmen aversion GDA. The main resul is ha he GDA uiliy funcion is a permanen ransformaion of he expeced uiliy funcion, in ha i amplifies risk premia a all invesmen horizons. Insead EZ uiliy is approximaely ransien ransformaion of expeced uiliy, implying very similar long-erm, bu differen shor-erm premia. Volailiy lieraure concenraes on invesigaion of wo-facor volailiy process, wih one facor being very persisen. In chaper I propose a differen paramerizaion of volailiy process ha includes his persisen componen as a sochasic cenral endency. The reparamerizaion is observaionally equivalen bu has compelling economic inerpreaion. I esimae he hisorical and risk-neural parameers of he model joinly using GMM wih he daa on realized volailiy and VIX volailiy index and reaing cenral endency as compleely unobservable. The main resul is ha on average he volailiy premium is indisinguishable from he premium on highly persisen shocks of he cenral endency. In chaper 3 I propose he exension of discree ime sochasic volailiy model in Darolles e al. 006 ha includes he leverage effec. There are several advanages of his model over commonly used coninuous-ime diffusions and discree-ime GARCH models. Firs, equiy risk premium and volailiy risk premium are known in closed form. Second, he model is robus o emporal aggregaion. Third, i has a well-known coninuous-ime limi. Moreover, hanks o exponenial affine form i is easy o compue opion prices in closed form hrough Fourier ransform. ii

3 Table of Conens Absrac ii Lis of Tables v Lis of Figures vi 1 Risk Premia: Shor and Long-erm Inroducion Asse Pricing Model Model Analysis Daa and esimaion Resuls Conclusion Pricing Cenral Tendency in Volailiy 39.1 Inroducion The Model Exac discreizaion Volailiy premia Esimaion Resuls Conclusion iii

4 3 Affine Opion Pricing Model in Discree Time Inroducion Discree ime sochasic volailiy wih affine dynamics and leverage effec Risk premia and opion prices GMM: general seup A Appendix for chaper A.1 Tables and Figures A. Proofs B Appendix for chaper 13 B.1 Tables and Figures B. Proofs C Appendix for chaper C.1 Tables and Figures C. Proofs Bibliography 187 iv

5 Lis of Tables 1.1 Momens in marix form A.1 Descripive saisics for annual consumpion growh, inflaion, riskfree rae, and marke excess reurn A. Means and sandard deviaions of he dividend growh raes for porfolios sored by book-o-marke BM, dividend yield DP, and size ME A.3 Benchmark model parameers A.4 Model implied reurn momens for benchmark parameers A.5 Transiion marix decomposiion. Model implied SDF ransiion marix S, modificaion marices K and G, SDF permanen componen Ŝ, principal eigenvecor φ, growh rae of SDF ρ S, growh rae of he pricing marix ρ G =.4415%, he difference beween he las Ŝ wo, uncondiional variance of SDF V S and of is permanen componen V. Firs row of he marix corresponds o he low curren consumpion growh. Firs column of he marix corresponds o he low fuure consumpion growh A.6 Risk premium in exreme cases of model parameers A.7 Risk premium in exreme cases of model parameers B.1 Descripive saisics for marke log reurns logr, opion-based volailiy index VIX, realized volailiy RV, and heir difference, volailiy risk premium VRP B. Esimaion resuls of he model wih sochasic rend. Third column parameers have he following inerpreaion: µ is he uncondiional mean of hisorical volailiy; κ σ and κ y are mean reversion speed parameers for volailiy and cenral endency, respecively, under hisorical measure; η σ and η y are insananeous diffusion parameers; λ σ and λ y are risk premia shif parameers necessary o obain risk-neural mean reversion speed parameers κ σ = κ σ λ σ η σ and κ y = κ y λ y η y. Risk-neural volailiy mean is given by µ = κσ κ y κ σ κ y µ v

6 Lis of Figures A.1 Sensiiviy o risk aversion parameer of risk premium. Firs and second columns depic shor erm and long horizons, respecively. Third column depics he difference beween shor and long-erm componens A. Sensiiviy o elasiciy of ineremporal subsiuion parameer of risk premium. Firs and second columns depic shor erm and long horizons, respecively. Third column depics he difference beween shor and long-erm componens A.3 Sensiiviy o generalized risk aversion parameer of risk premium. Firs and second columns depic shor erm and long horizons, respecively. Third column depics he difference beween shor and long-erm componens A.4 Sensiiviy o kappa parameer of risk premium. Firs and second columns depic shor erm and long horizons, respecively. Third column depics he difference beween shor and long-erm componens A.5 Sensiiviy o probabiliy ransiion parameer of risk premium. Firs and second columns depic shor erm and long horizons, respecively. Third column depics he difference beween shor and long-erm componens A.6 Sensiiviy o mean consumpion growh rae parameer of risk premium. Firs and second columns depic shor erm and long horizons, respecively. Third column depics he difference beween shor and long-erm componens A.7 Sensiiviy o consumpion growh sandard deviaion parameer of risk premium. Firs and second columns depic shor erm and long horizons, respecively. Third column depics he difference beween shor and long-erm componens A.8 Time series of marke marke risk premium, reurn, and risk-free rae over holding period A.9 Time series of equiy risk premium, reurn, and is difference wih marke risk premium. Dividend growh mean is µ e = %, and sandard deviaion is δ e = 5%.114 A.10 Time series of equiy risk premium, reurn, and is difference wih marke risk premium. Dividend growh mean is µ e = 0%, and sandard deviaion is δ e = 10% B.1 Daily S&P500 index SPX and marke log reurn logr B. Daily opion-based volailiy index VIX, and realized volailiy RV vi

7 B.3 Auocorrelaion funcion for opion-based volailiy index VIX, realized volailiy RV, and log marke reurn logr B.4 Implied sandard deviaions of daily volailiy VV, cenral endency VY, and heir difference VV-Y. Implied 95% confidence inervals are given by dashed lines B.5 Implied correlaions beween volailiy V, cenral endency Y, and heir difference T. Implied 95% confidence inervals are given by dashed lines B.6 Implied auocorrelaions of volailiy V, cenral endency Y, and heir difference T. Implied 95% confidence inervals are given by dashed lines B.7 Implied means of daily volailiy premium VP, cenral endency premium CP, and heir difference ransiory premium TP. Implied 95% confidence inervals are given by dashed lines B.8 Implied sandard deviaions of daily volailiy premium VP, cenral endency premium CP, and heir difference ransiory premium TP. Implied 95% confidence inervals are given by dashed lines C.1 Volailiy ARG1 and correlaion wih disurbance erm, c = 0.01, δ = 1.047, ρ = 0.96, and φ = C. Parameer φ as a funcion of arge correlaion φ 0. Parameer values: c = 0.01, δ = 1.047, and ρ = C.3 Implied volailiy smile across differen leverage parameers. Parameer values are T = 10/365, Eσ = c/ 1 ρ =.5/N, N = 0 365, δ = and ρ = vii

8 Chaper 1 Risk Premia: Shor and Long-erm

9 1.1 Inroducion The financial economics lieraure on risk-reurn rade-off and risk compensaion is voluminous and growing. Mos of he heoreical developmens in his area focus on shor or even insananeous ime frames. Very lile is known abou risk and is compensaion for agens who inves over a long ime span. The main ineres of his paper is he premium on he dividend flows wih an infinie horizon. In numerous aemps o resolve conflics beween heory and empirics, asse pricing models have been exended o include a variey of differen preference srucures. As Hansen 008 shows many of hese uiliies have only ransien implicaions and make no difference in pricing asses wih long mauriy. In his paper I consider a discree-sae economy and consruc an asse pricing model for he valuaion of consumpion and dividend cash flows wih shor and long mauriies which generalizes he model of Mehra and Presco I use he mehodology of semigroup asse pricing heory Hansen and Scheinkman, in order o consruc ransiion marices of sochasic discouning and pricing. I consider hree uiliy funcions: expeced uiliy EU, Epsein and Zin 1989 EZ, and generalized disappoinmen aversion GDA proposed by Rouledge and Zin 004. The main resul of he paper is ha he GDA uiliy funcion amplifies risk premia a all invesmen horizons in comparison o expeced uiliy funcion. Insead EZ uiliy delivers very similar long-erm premia, bu differen shor-erm premia compared o EU. Hence, we say ha GDA is a permanen ransformaion of EU, while EZ is a ransien ransformaion of EU. 3 The main feaure of GDA uiliy ha produces subsanially differen risk premia for any horizon is is asymmeric response o differen economic oucomes hrough he implied skewed risk-neural probabiliies. Long-erm reurns are also an inegral par of shor-erm reurns. Their difference may be 1 See Garman 1985; Anderson e al. 003; Ai-Sahalia e al. 004 for he genesis of he semigroup asse pricing heory. Disappoinmen aversion concep is developed by Gul 1991 and subsequenly generalized by Bonomo and Garcia 1994 and Rouledge and Zin 004. Also see Backus e al. 004 for oher preference srucures. 3 Oher ransien modificaions of expeced uiliy are seen in Consaninides 1990; Campbell and Cochrane 1999; Heuvel 008; Sanos and Veronesi 010.

10 inerpreed as a erm premium. 4 If a shor-erm risk premium is smaller han is long-erm counerpar, hen he erm premium is posiive. This means ha an invesor is rewarded for holding an asse for a long period of ime. The long-erm reurn does no depend on he curren sae of economy and hence is a consan. So all he shor-erm variaions in he reurns are due o flucuaions in erm premia. In his paper I show ha a consumpion based model can generae posiive erm premium for any holding period and any dividend sream. Moreover, his suggess ha in order o undersand shor-erm asse pricing phenomena one should no overlook long-erm componen of he model. The paper is relaed o he work by Alvarez and Jermann 005 and Hansen and Scheinkman 009 ha propose muliplicaive decomposiion of sochasic discoun facors SDF ino exponenial growh, permanen componen, and ransien componen. Exponenial growh deermines long-erm reurns. The permanen componen alers he probabiliy disribuion and incorporaes he preference srucure ino risk-neural probabiliies. This componen is he main facor in he pricing of he long-erm risk. Transien componen only affecs pricing a he shor invesmen horizons. Bansal and Yaron 004 use his shor-erm feaure of EZ uiliy o build a consumpion based model and solve an equiy premium puzzle a he shor invesmen horizon. Bu in he long run many asse differences disappear and leave an invesor s porfolio exposed o a few defining rends similar for many uiliy specificaions. I documen he parameric sensiiviy of he model for shor- and long-erm risk premia. In paricular, I show ha he EZ and GDA implied long-erm risk premia are no a funcion of he elasiciy of ineremporal subsiuion. Hansen e al. 008 price a cross-secion of sock reurns a long horizons and find ha he elasiciy of ineremporal subsiuion EIS is no a significan facor in deermining long-erm porfolio reurns. In his paper I generalize his finding o a framework in which he SDF is no approximaed around uni EIS. Moreover, I documen ha he weak dependence on EIS perains o more general recursive GDA uiliy. An asse reurn is he raio of he fuure value of he asse and is curren price. Consruced 4 See Daniel and Marshall 1997; Lemke and Werner 009; Berg 010; Binsbergen e al. 010 for furher discussion on he erm srucure of risk premia. 3

11 as such he one period, or shor-erm, reurn may be used o consruc he long-erm reurn as he geomeric average of consecuive shor-erm reurns. Hansen and Scheinkman 009 consruc a family of operaors ha maps fuure saionary payoffs o is curren prices. Then he infinie horizon premium is equal o he combinaion of long-erm growh raes of hese operaors. I assume ha consumpion and dividend growh raes can ake only a finie se of values. Hence, an economy sae is represened by a pair of consumpion and dividend growh raes. The ransiion beween saes of he economy is governed by a single probabiliy ransiion marix. The consumpion/dividend process is hus a Markov chain of order one. This framework leads o a convenien represenaion of operaors as ransiion marices. This reduces all calculaions o marix algebra. The final oucome is ha he long-erm reurn is simply he larges real eigenvalue of he corresponding pricing ransiion marix. The discree ime finie sae model considered here has wo ighly linked advanages. Firs, he implicaions are very robus o he number of saes of he economy. Second, in he wo-sae model all he parameers have a clear and inuiive inerpreaion in erms of preference parameers, consumpion persisence, means and variances of cash flows. Here I show ha in discree ime model he EZ modificaion of expeced uiliy is no exacly ransien. This goes in conras o he coninuous ime model of Hansen 008 where he shows ha EZ modificaion only affecs ransien componen. In my model i modifies risk-neural probabiliies, or permanen componen, as well. Empirical analysis confirms his finding alhough shows ha he deviaion is negligible. I find ha he GDA preference srucure proposed by Rouledge and Zin 004 is no a ransien modificaion of expeced uiliy. Invesors wih such preferences are disappoined disproporionaely more by a recession han hey are pleased wih an equally posiive economic oucome. This aiude oward risk shifs risk-neural probabiliies oward he bad oucomes of he economy hus alering he permanen componen of he SDF. The risk-neural probabiliy shif reduces asse prices, and i increases expeced risky reurns. Bonomo e al. 010 show ha his model amplifies reurns a he shor horizon. Here I exend his conclusion o he 4

12 long horizon. Empirical analysis suppors his finding. The paper is organized as follows. Secion 1. inroduces basic definiions and resuls regarding Markov chains in discree space. In he same secion I ouline asympoic resuls for discree space ransiion marices. These marices applied o familiar financial indicaors provide a very convenien oolbox for compuing and comparing asse reurns and risk premia for differen asses, differen uiliy funcions, and invesmen horizons. Secion 1.3 presens heoreical analysis of he model and shows ha GDA modificaion of he EU and EZ uiliies has a permanen effec and amplifies premia a any horizon. Secion 1.4 gives a descripion of he daa and esimaion mehod. Secion 1.5 presens he resuls of numerical soluion of he model 5 and explains in deails he implicaions of he hree uiliy models for reurns, risk premia, is permanen and ransien componens. Secion 1.6 concludes. 5 The numerical sudy considers for simpliciy of exposiion only wo sae economy. However, addiional numerical work immediaely expandable o arbirary number of saes is available upon reques. I shows ha he main conclusions are robus o he number of saes. 5

13 1. Asse Pricing Model 1..1 Consumpion and dividend dynamics In his secion I describe he join dynamics of consumpion and asse dividends. Then I show how o price any asse given is dividend sream and probabiliy disribuion. Assumpion 1. Join consumpion and dividend dynamics Aggregae consumpion sream evolves according o C +1 = C x +1. The dividend sream evolves according o D +1 = D y +1. Each sae of he economy is characerized by a pair of consumpion and dividend growh raes, z = x, y, aking values in a finie se, {λ 1, ν 1,..., λ n, ν n } wih z i = λ i, ν i. The finie sae probabiliy ransiion n n marix P = p ij n i,j=1 now describes evoluion of a wo-dimensional random variable, wih elemens corresponding o consumpion and dividends. So, is definiion: p i,j = P x +1, y +1 = λ j, ν j x, y = λ i, ν i. For fuure convenience define he vecor of consumpion growh raes λ = λ 1,..., λ n T and dividend growh raes ν = ν 1,..., ν n T. Along wih ha define wo diagonal marices: Λ = diag λ, and Θ = diag ν. Wih his firs order Markov srucure i is raher easy o redefine condiional expecaion hrough marix language. For any funcion ψ of a sae we can define condiional expecaion operaor in marix language as E ψ z i =E ψ z +1 z = z i n = P z +1 = z j z = z i ψ z j j=1 n = p ij ψ z j = Pψ z i. j=1 Addiionally, if I denoe he vecor ψ = ψ z 1,..., ψ z n T, hen is condiional expecaion can be fully described by E ψ = Pψ. Now recall ha he soluion, if i exiss, of π = P T π is 6

14 he marginal probabiliy disribuion of he sae vecor, π i = P z = z i. Using his definiion one could also define uncondiional expecaion Eψ = π T ψ. Noe ha he above funcion ψ is ime homogeneous, ha is is values only depend on curren sae of he naure. Now suppose ha we have a marix Ψ = {Ψ z i, z j } ha describes he value of random variable as a funcion of a sae a ime + 1, z j, and sae s curren sae a ime, z i. Obvious example is he asse reurn. Then is expecaion condiional on he curren sae is n E Ψ z i, z +1 = p ij Ψ z i, z j. j=1 The column vecor of hese condiional expecaions indexed by condiioning sae value z i is convenienly wrien using componen-wise Hadamard produc of marices: E Ψ = P Ψ 1 n. Following his idea I summarize expressions for firs and second momens in he following able: Mean Variance Condiional E Ψ = P Ψ 1 n V Ψ = P Ψ 1 n P Ψ 1 n Uncondiional EΨ = π T P Ψ 1 n VΨ = EV Ψ + VE Ψ Table 1.1: Momens in marix form In he above able he componens of uncondiional variance are VE Ψ = π T E Ψ π T E Ψ, and EV Ψ = π T V Ψ. Consider he cash flow decomposed ino a growh componen y +1 and a saionary componen ψ +1 = ψ z j. Then is curren condiional expecaion is compued as n p ij ν j ψ z j = PΘψ z i = Gψ z i, j=1 7

15 or in marix noaion Payoff = Gψ. The canonical asse pricing model is P = E S,+1 ψ +1, where P is he curren value of he fuure payoff ψ +1, and S,+1 is he Sochasic Discoun Facor SDF. Expecaion is aken wih respec o he curren informaion se. In he finie sae economy, n P i = p ij s ij ψ z j = Sψ z i, or P = Sψ. j=1 This equaion may be used o price he uni riskless payoff, P f = S1 n. Taking he reciprocal of he price, R f 1 = S1 n 1, and ransforming o log scale I obain generic formula for he log risk-free reurn: log R f 1 = log S1 n. Now, he curren price of a fuure growing payoff is n P i = p ij s ij ν j ψ z j = SΞψ z i = Qψ z i, or P = Qψ. j=1 This leads o he definiion of one period equiy reurn in log scale: log R e 1 = log Gψ log Qψ. The naural exension of he above definiions is pricing of cash flows over muliple periods. In finie space his exension reduces o aking marix powers. Wih a sligh abuse of noaion his saemen is proven below using he law of ieraed expecaions: P,+h =E S,+h ψ +h = E S,+h 1 E +h 1 S +h 1,+h ψ +h =E S,+h 1 Sψ = E S,+h S ψ =... = S h ψ. 8

16 Applying his resul o asse reurns he average log equiy reurn a horizon h is log R e h = 1 h log G h ψ log Q h ψ, and he risk-free rae of reurn is log R f h = 1 h log S h 1 n. The equiy risk premium is herefore defined as RP e h = log R e h log R f h. I urns ou ha he limi of his expression for an infinie payoff horizon has precise mahemaical expression. Defining ρ M as a principal eigenvalue, he eigenvalue of log M wih he larges real par, of a marix M i is shown ha RP e 1 = lim log G h ψ log Q h ψ + log S h 1 n h h =ρ G ρ Q + ρ S. This resul does no exhaus he imporance of principal eigenvalues for asse pricing. Following Hansen and Scheinkman 009 I show ha hey play an imporan role in decomposiion of asse reurns a any payoff horizon. I urns ha under cerain regulariy condiions a marix M h ha may sand for any of G h, Q h, or S h is decomposed as follows: M h = exp ρh Φ ˆM h Φ 1 = exp ρh M h, where ρ is already inroduced specral radius of he marix, Φ = diag φ is he diagonal marix, φ is he corresponding n 1 eigenvecor. More specifically, ρ and φ are he soluions 9

17 of M h φ = exp ρh φ. Using his decomposiion I can wrie he following: 1 h log M hψ = ρ + 1 h log M h ψ. Applying his o he risk-free rae I obain he following decomposiion: log R f h = log Rf + log R f h, where I denoed log R f = ρ S, and log R f h = 1 h log Sh 1 n. Similar decomposiion works for equiy reurn as well: log R e h = log R e + log R e h, wih log R e = ρ G ρ Q, and log R e h = 1 h log Gh ψ log Qh ψ. Finally, he difference beween he wo above, equiy risk premium may be wrien as RP e h = log R e h log R f h = RP e T P e h, 1.1 where risk premium on an infiniely lived asse is RP e = log R e log R f, and he second erm in premium decomposiion is T P e h = log R f h log R e h. In he above expressions I simply add ilde o he noaion o specify ime dependen componen of a reurn. Permanen componen has infiniy in is subscrip. The ime dependen erm in 10

18 risk premium decomposiion has a naural inerpreaion of he erm premium, denoed by T Ph e. This value says how much does an invesor gains/looses if she holds an asse for only h periods raher han keeping i indefiniely longer. If say T P e 1 is negaive, hen one period reurn is larger han on he infiniely held asse, so he invesor is beer off by limiing her invesmens in his asse o only shor erm. On he oher hand, if T P e 1 is posiive, hen he invesor is beer off by commiing o he longer holding period. In he limi his value becomes zero, and he risk premium converges o is long-erm counerpar, lim T P h e = 0, h lim RP h e = RP. e h The same conclusion holds for each individual reurn in absolue erms. 1.. Pricing wih expeced uiliy Define ineremporal expeced uiliy funcion or simply expeced uiliy furher in he ex as U 0 = E 0 β C1 γ 1, 1 γ =0 where β is subjecive discoun facor and a is a coefficien of relaive risk aversion RRA. Sochasic discoun facor SDF can be wrien as S,+1 = β γ C+1. C Take for he purpose of illusraion β = 1 and γ = 0. This combinaion of parameers implies linear uiliy funcion U 0 = =0 E 0 C, and a consan sochasic discoun facor. This is exacly he case of a risk-neural invesor. The Euler equaion ha prices nex period random payoff ψ +1 for his paricular SDF is P = E β C+1 C γ ψ

19 Given Assumpion 1, we can wrie he price as P i = β n j=1 p ij λ γ j ψ j, or P = βpλ γ ψ. This expression shows ha he power uiliy model is characerized by he following marix SDF: S X = βpλ γ Pricing wih Epsein-Zin uiliy In his secion I obain pricing operaors implied by a paricular case of Epsein-Zin uiliy funcion. Define Kreps-Poreus special case of Epsein-Zin EZ hereafer recursive uiliy funcion see Epsein and Zin, 1989; Kreps and Poreus, 1978: V = C σ + β R V σ σ, where risk adjusmen or cerainy equivalen is given by R V +1 = E V 1 γ γ. Some auhors use anoher paramerizaion such as ρ = 1 1 σ and a = 1 γ. Such definiions are useful for reconciliaion of resuls from differen sources. SDF implied by such a uiliy funcion can be wrien as S,+1 = β C+1 C 1 1 σ V +1 σ γ. R V +1 Compare his SDF wih corresponding expression for power uiliy funcion. Noe ha if one imposes he resricion ha risk aversion parameer is he reciprocal of ineremporal subsiuion coefficien, γ = 1 σ, hen Epsein-Zin SDF reduces o he one implied by power 1

20 uiliy funcion. As i was shown by Epsein and Zin 1989 EZ uiliy implies he following SDF in erms of well known and measurable quaniies: S,+1 = β C+1 C 1 γ 1 σ βr m,+1 1 γσ σ 1, hence he SDF in marix form is S Z = βpλ 1 γ 1 σ βr m 1 γσ σ 1 = SX βr m 1 γσ Λ 1 σ 1 = S X K, 1.3 where R,+1 m is he reurn on marke porfolio. As i is commonly used in he lieraure see e.g. Garcia and Renaul, 1998, marke payoff is he oal endowmen of he economy, C. Hence, he reurn is expressed as R,+1 m = P +1 m +C +1 P m, where P m is he price of marke porfolio. In he curren model consumpion growh raes are known bu marke reurns necessary for asse pricing are no. Clearly, he asse price compued using EZ uiliy coincides wih he one from expeced uiliy wih resricion ha 1/σ = γ. As a confirmaion of noe ha SDF derived from EZ uiliy 1.3 reduces o SDF in expeced uiliy 1.. Now ake β = 1 and γ = 0. This exreme case implies ha he SDF is no consan and equal o S,+1 = R+1 m C C +1 1 σ 1. Noe ha γ = 0 corresponds o risk-neural agen. Even wih such an agen, he SDF remains random and will cerainly covary wih consumpion equiy and even wih some dividend sreams. Thus, he risk premium is no expeced o be zero even for a risk-neural agen. 13

21 1..4 Pricing wih GDA uiliy The SDF for Generalized Disappoinmen Aversion GDA developed by Bonomo and Garcia 1994; Bonomo e al. 009, 010 S,+1 =β C+1 C 1 σ V +1 1 σ γ R V +1 V α 1 I R V +1 < κ 1 + κ 1 γ α 1 E I V+1, R V +1 < κ where parameer α is inerpreed as disappoinmen aversion, and κ as disappoinmen hreshold. They show ha he SDF may be rewrien as S,+1 = h 1 γ +1 R m α 1 I h,+1 < κ κ 1 γ α 1 E I h,+1 < κ, where h,+1 = C+1 C 1 1 σ βr m,+1 σ σ 1. In marix noaion ha would look as follows: 1 + α 1 I H < κ S G = S Z 1 + κ 1 γ α 1 E I H < κ 1 T n = S Z G, where condiional expecaion is E I H < κ = P I H < κ 1 n, and H = βr m σ σ 1 Λ 1 1 σ. Noe ha whenever α = 1 we are back o he case of EZ uiliy. In order o undersand he inner workings of his uiliy funcion le me ake γ = 0, σ =, κ = β = 1, and condiional on he curren sae of he economy marke reurn is equally likely o be above or below one. This combinaion of preference parameers reduces he SDF o he 14

22 following expression: S,+1 = 1 + α 1 I R,+1 m < α 1 E I R,+1 m < 1 = 1, if R,+1 m 1 α > 1, if R,+1 m < α / Now everyhing depends on he marke reurn. This implies only wo possible values for he SDF: / 1 + α and α/ 1 + α. The raio beween hese wo is α. As long as his parameer is chosen o be greaer han one, he raio beween wo possible SDF values is greaer han one. Moreover, he larger SDF value corresponds o he negaive oucome of he economy, when consumpion equiy looses in value. Hence, he SDF becomes amplified for bad oucomes. This creaes negaive covariance beween SDF and risky reurn. In addiion, even when oher uiliies imply consan SDF, GDA uiliy implies ruly sochasic discoun facor General soluion Nex, price of an asse can be wrien as condiional expecaion of discouned fuure payoff price plus dividend: P = E S,+1 P +1 + D Since he soluion is ime invarian, we can wrie he price indexed by sae and condiional on iniial dividend value d as n P d, i = s ij P dν j, j + dν j. j=1 Since price is homogeneous of degree one in d, he soluion can be wrien as P d, i = w i d. Here w i is a sae dependen bu ime homogeneous price-dividend raio. This implies he following n equaions wih n unknowns: n w i = s ij ν j w j + 1. j=1 15

23 Or wih vecor noaion w = w 1,..., w n T, hese equaions can be wrien as w = Q w + 1 n, I is sraighforward o ge he soluion for he price-dividend raio vecor: w = Q 1 I n 1 1n. 1.6 This formula is o be used for any uiliy funcion and any asse considered in his paper. Equivalen sysem of equaions ha is more suiable for numerical soluion mehods may be wrien as 1 n = S R e 1 n, where R e = w 1 w + 1 n T Θ. In order o solve he model wih EZ uiliy i is firs necessary o find he expression for he marke reurn R m. Provided ha he soluion is ime invarian and homogeneous of degree one, hen marke reurn may be wrien in erms of consumpion growh raes λ j and marke price-dividend raio w mj : R m ij = P λ jc, j + λ j c P c, i = λ j w mj + 1 w mi, or in marix noaion R m = w 1 m w m + 1 n T Λ. Subsiue his expression in order o ge explici expression for marix SDF in EZ case: S Z =βpλ 1 a 1 σ βr m 1 aσ σ 1 =βpλ a βwm 1 w m + 1 n T 1 aσ σ =S X K. 16

24 Given he SDF marix i is he following sysem of equaions ha is o be solved on he firs sep: 1 n = S R m 1 n. This generic equaion is o be used o solve for price-dividend raio for boh EZ and GDA uiliy funcions 6. The second sep uses 1.6 o solve for price-dividend raio of a generic asse. 6 This sysem of n non-linear equaions wih n unknowns does no have a closed form soluion so numerical mehods provided by MaLab non-linear equaion solver are used in empirical secion. 17

25 1.3 Model Analysis In his secion I presen he analyical invesigaion of he model. I inen o show ha in discree ime he EZ uiliy which is a modificaion of expeced uiliy is very close o being ransien. The disance depends on how large and how volaile marke price-dividend raios. I will also show ha he GDA uiliy modificaion is permanen in he sense ha i modifies riskneural probabiliy disribuion. This will increase SDF variance and evenually he disance beween he long-erm risky and riskless reurns, he risk premium Permanen and Transien Modificaions As i was highlighed in Secion 1. he muliplicaive decomposiion of an SDF ransiion marix akes he form S = exp ρ Φ Ŝ Φ where ρ is he larges real eigenvalue of he marix S, Φ = diag φ is he diagonal marix, φ is he corresponding n 1 eigenvecor. Here Ŝ plays a role of probabiliy ransiion marix bu wih risk-neural probabiliies ha ake ino accoun preference srucure and aiudes oward risk. In erminology of Hansen and Scheinkman 009 his marix is a permanen componen. This applies for he following inuiive reason. Suppose we are ineresed in pricing dividend flow over cerain horizon h. For ha we have o ake he power of he pricing ransiion marix and muliply i by he vecor of saionary payoffs. So he h s power of a generic ransiion marix is S h = exp ρh Φ Ŝh Φ 1. Noice ha cumulaive muliplicaion of he marix cancels ou Φ and Φ 1 in he middle so ha he only accumulaed componens are exponenial growh and Ŝ. Hence, any change in his marix propagaes o any horizon. A he same ime, his jusifies he erminology for Φ, ransien componen. 18

26 Now consider ha we have a modificaion of he SDF ransiion marix ha fis ino he following form: S = W 1 S W, 1.9 where W is a diagonal marix. Using his modificaion muliply he original decomposiion 1.8 from he righ by W and by is inverse on he lef: W 1 SW = exp ρ W 1 Φ Ŝ Φ 1 W. The lef hand side should immediaely be recognized as he modified marix S. Bu does he righ-hand side remains he muliplicaive decomposiion? The answer is yes due o he same inuiion I oulined above: S h = exp ρh W 1 Φ Ŝh Φ 1 W. The diagonal marices W 1 Φ and Φ 1 W are no compounded, while exponenial growh and Ŝ are. Even more imporan ha growh and permanen componen are unalered. All of he modificaion is gone o he ransien componen. Such a modificaion of a ransiion marix may be called ransien. From he definiion of EZ sochasic discoun facor marix in 1.7 i follows ha he modificaion of expeced uiliy SDF akes he following form: S Z = W θ S X W + I θ, where θ = 1 aσ σ 1 and W is he diagonal marix of marke price-dividend raios. Suppose ha he decomposiion of S X is he same as in 1.8. Once again, our evenual arge is pricing of he long daed asses. So, for a sar ake he second power of he SDF marix: S Z = W θ S X I + W 1 θ SX W + I θ. 19

27 Clearly, he erm in he middle does no reduce o he ideniy. On he oher hand, if he diagonal elemens of W, marke price-dividend raios, are large enough, and power θ is no oo large, hen he middle erm may be considered relaively small. Moreover, if he diagonal elemens are no significanly differen from each oher, hen he major effec is only in scaling. So I may wrie approximae relaionship S h Z exp ρ Z h ΦW θ Ŝh X W + I θ Φ 1. I expec ha he growh/decay ρ Z of EZ ransiion marix will be differen han ha of expeced uiliy since i has o compensae he growh/decay of he marix I + W 1 θ. So he EZ modificaion is approximaely ransien in discree ime. Hansen e al. 008 show ha in coninuous ime he same modificaion is exacly ransien. Here, he magniude of he ransien effec depends on he magniude and variaion of marke price-dividend raios which are he enries of marix W. The smaller and more variable hey are he closer he modificaion o being ransien. We see ha he growh rae of EZ discoun facor may be differen from he one implied by he expeced uiliy. Hence, long-erm expeced reurns will be differen in boh model. I inen o show ha alhough long-erm reurns are differen, he long-erm risk premia are he same. Consider he pricing ransiion marix of he expeced uiliy, Q X = S X Λ, and using he decomposiion of SDF marix wrie Q X = S X Λ = exp ρ X Φ X ŜXΛ Φ 1 X. Naurally, he marix ŜXΛ does no posses he propery of probabiliy ransiion marix. So, decompose i as Ŝ X Λ = exp ρ X Φ X S X and plug his decomposiion back o pricing marix: Φ 1 X, Q X = exp ρ X + ρ X Φ X ΦX S X Φ 1 X Φ 1 X. 0

28 Similarly, decompose he pricing marix of EZ uiliy funcion: Q Z exp ρ Z + ρ X Φ X ΦX W θ S X W + I θ Φ 1 X Φ 1 X. Noice ha boh pricing marices, Q X and Q Z, have exacly he same modificaion ρ X in he growh rae. In his noaion he long-erm risk-free rae is defined as ρ X and ρ Z for EU and EZ models, respecively. Long-erm expeced asse reurn is deermined by dividend growh marix which is unalered, and by pricing marix growh raes ρ X + ρ X and ρ Z + ρ X for each model. This leaves he difference beween risky and riskless reurns equal o ρ X for each of he wo models. In oher words, his proves ha he long-erm risk premia are very close o each oher in EU and EZ models. This conclusion is solely due o he fac ha he permanen componen, or risk-neural probabiliy ransiion marix, is shared by boh models. Consider he modificaion of SDF implied by he GDA uiliy funcion. Recall ha his modificaion akes he form 1 + α 1 I H < κ S G = S Z 1 + κ 1 γ α 1 E I H < κ 1 T n = S Z G, where condiional expecaion is E I H < κ = P I H < κ 1 n, and H = βr m σ σ 1 Λ 1 1 σ. Clearly, his modificaion may no be represened in he symmeric form in 1.9 or even he one implied by EZ uiliy. Take for example he exreme case where risk aversion is zero, γ = 0, elasiciy of ineremporal subsiuion is infinie, σ =, discoun rae and disappoinmen hreshold are one, κ = β = 1, and condiional on he curren sae of he economy marke 1

29 reurn is equally likely o be above or below one. This resricion produces SDF as in 1.4: S GDA,+1 = G = 1, if R,+1 m 1 α > 1, if R,+1 m < α / Noe ha for a given curren sae of he economy he SDF akes only wo values, 1 and α, up o a consan 1+α. In order for his modificaion o be symmeric in he sense of 1.9, he SDF has o ake a leas hree values one of which should be 1/α, which is no he case here. Hence, he modificaion is no symmeric and does no admi he form in 1.9. So, he decomposiion of he SDF is S h G = exp ρ G h Φ G Ŝh G Φ 1 G. Each componen in his decomposiion is poenially differen from is counerpar of oher uiliies. Mos imporanly, he permanen componen, he risk-neural probabiliy ransiion marix, is modified. So he GDA uiliy modificaion may be considered permanen. Recall ha GDA is consruced in such a way ha i is amplified when he fuure oucome is perceived o be bad for an invesor. This implies ha he risk-neural probabiliies will be skewed owards possible economic downurns. Ineviably such il in probabiliies will amplify he negaive covariance of he SDF and risky reurns. This in urn will lead o he increased disance beween growh raes in risky and riskless reurns. Hence, he long-erm risk premium is expeced o be amplified along wih he shor one Risk Premia Approximaion In his secion I presen an approximae analyical soluion in he simples possible case of he wo-sae economy. This soluion will give me a general idea of differences beween shor and long horizon risk compensaion. Assume ha we are in he wo-sae world wih consumpion going up or down. Now hink of an asse wih a pro- or counercyclical dividend flow. Pro

30 cyclical dividends would increase wih consumpion increase alhough a a differen pace. This se up is a generalizaion of he model developed by Mehra and Presco Assumpion. Consumpion and dividend dynamics Le consumpion evolve wih growh raes λ 1, = 1 + µ c ± δ c wih λ 1 λ. Growh raes of generic asse dividends in corresponding saes are ν 1, = 1 + µ e ± δ e wih ν 1 ν. Here µ c, µ e, δ c, δ e 0. Moreover, ransiion probabiliies are p 11 = p = p, p 1 = p 1 = 1 p wih p 0, 1. The relaion beween ν 1 and ν would deermine if an asse is pro cyclical ν 1 > ν or counercyclical ν 1 < ν. I is easy o show ha Corr λ, ν = sgn ν 1 ν boh condiional and uncondiional. Also noe ha his does no imply perfec absolue correlaion beween marke and asse reurns. In his simple model he firs and second momens of consumpion growh are convenienly mapped o model parameers. Firs, saionary mean and variance of consumpion growh rae: Ex = 1 λ 1 + λ = 1 + µ c, V x = 1 4 λ 1 λ = δc. Condiional means: E x x 1 = λ 1 =pλ p λ = 1 + µ c + δ c p 1, E x x 1 = λ = 1 p λ 1 + pλ = 1 + µ c δ c p 1. Firs order auocorrelaion: Corr x, x 1 = Cov x, x 1 V x = p 1. Dividend growh y momens are analogous. These derivaions show ha in he wo-sae economy as in Assumpion he growh rae momens have convenien one-o-one mapping from model parameers µ, δ, and p. The firs parameer becomes mean growh rae, he second is is sandard deviaion, and he las one is a posiive linear funcion of auocorrelaion. Such a connecion beween model parameers and growh rae momens subsanially simplifies he subsequen analysis. Noe ha in a model wih a greaer number of saes, non-resriced 3

31 growh raes, and ransiion probabiliies, here does no exis a on-o-one mapping beween numerous parameers, mean, variance, and auocorrelaion. This jusifies he use of he wosae model Shor-erm risk premium Firs of all denoe ˆλ = λ 1 /λ and ˆν = ν 1 /ν. In Appendix A. I show ha in proximiy of a = 0, or p = 0, or p = 1, or ˆν, ˆλ = 1 he expeced shor-erm one period risk premium implied by GDA uiliy funcion is RPG1 e p 1 p ˆν w w a ˆλ g1 + g aσ 1 wm1 + 1 σ 1 1. g 11 g 1 w m + 1 As we know, EZ uiliy is a specific case of GDA given disappoinmen aversion is zero, α = 0. Applying his resricion o he above expression I obain RPK1 e p 1 p ˆν w w a ˆλ 1 + aσ 1 wm1 + 1 σ 1 1. w m + 1 Similarly, expeced uiliy is only a specific case of EZ if risk aversion is zero, a = 0. This resricion furher simplifies he expression for he risk premium: RPX1 e ap 1 p ˆλ ˆν w w Noe ha all hree expressions are increasing funcions of he risk aversion a. Bu a he same ime, if one hinks of a risk-neural agen, ha is a = 0, he only risk premium ha unambiguously becomes zero is he one implied by expeced uiliy. The oher wo are non-zero in general. Firs look a he EZ risk premium. The addiional erm differeniaing EZ from is specific case depends on he raio of marke price-dividend raios. As long as he wo do no differ much across economy saes, he raio will be close o one, and he risk premium close o zero. The siuaion wih GDA risk premium is poenially quie differen. Using he definiion 4

32 of GDA sochasic discoun facor, g 1 = 1 + α 1 1 I h 1 < κ g α 1 1 I h 11 < κ, and g = 1 + α 1 1 I h < κ g α 1 1 I h 1 < κ. Given cerain combinaions of parameers ha sufficienly differeniae GDA uiliy from EZ, i is likely o see α or 1/α as he values for he above raios. So for disappoinmen aversion parameer α sufficienly below one hese raios are also very far from one. Hence, I expec ha he risk premium implied by GDA uiliy is very far from zero even for risk-neural agen in he sense of EZ and expeced uiliies. Now look a he effecs of oher parameers. Probabiliy parameer p is inerpreed as persisence of growh raes or equivalenly as a measure of predicabiliy. Values of p far from 50% are a sign of large absolue auocorrelaion and herefore greaer predicabiliy. So a leas he firs order effec of p being close o 50% or increased uncerainy is an increase in risk compensaion. Also noe ha risk premium is an increasing funcion of percen raio beween consumpion growh raes ˆλ 1. Nex, observe ha all hree expressions have a common muliplier, p 1 p ˆν w w Alhough analyically i is he same, numerically i will be differen as long as price-dividend raios are differen for each model. Analyically he las erm in his expression is inerpreed as a percenage difference beween fuure asse payoff in he good sae, ν 1 w 1 + 1, and in he bad sae ν w + 1. This raio is in direc connecion wih he range of payoff flucuaions. The wider payoff flucuaions he larger he risk premium. Having he above expressions i is quie naural o resric hem o he case of consumpion equiy o obain he expression for marke risk premium. This resricion is wrien as ν 1 = λ 1 and ν = λ, which implies ˆν = ˆλ. This leads o he following raio beween equiy and marke 5

33 risk premiums: The erm RP1 e RP1 m ˆν w w / ˆλ w m1 + 1 w m ˆν w 1+1 w +1 1 should be hough of as he percen raio beween asse payoffs in differen saes of he economy. The denominaor is he percen raio beween fuure consumpion in differen saes of he economy. Consequenly, his consruc is immediaely recognized as he elasiciy of subsiuion beween equiy and marke payoffs. Naurally, if dividends grow a he same raes as consumpion, his raio will be equal o one. We will see laer ha analogous raio of long-erm risk premia is subsanially simpler and only depends on growh raes of consumpion and dividends Long-erm risk premium In Secion A. I show ha in proximiy of a = 0, or p = 0, or p = 1, or ˆν, ˆλ = 1 he expeced long-erm risk premium implied by GDA uiliy funcion is RPG e p ˆν 1 a ˆλ 1 + kg /g Similarly o he previous secion I se α = 1 in order o obain EZ implied risk premium: RPK e p ˆν 1 a ˆλ 1 + k 1. And he las one, expeced uiliy risk premium, is obained by seing a = 1/σ: RPX e ap ˆν 1 ˆλ 1. Firs of all noe ha again he risk premia are increasing funcions of risk aversion parameer a. Bu, in conras o expeced uiliy where risk neural agen receives no compensaion, boh GDA and EZ preferences imply non-zero compensaion in general. As I argued in he previous secion parameer k is very close o zero as long as price-dividend raios do no vary subsanially across economy saes. A he same ime he raio g /g 11 is likely o be equal 6

34 o 1 + κ 1 a α 1 1 E I h < κ 1 + κ 1 a α 1 1 E 1 I h 1 < κ. I is much more likely ha an agen will be disappoined while currenly being in he good sae of he economy. So i is more realisic o assume ha his raio is subsanially greaer han one. This suggess ha GDA implied long-erm risk premium will be much differen from zero for risk-neural agen as measured by a. Nex noe ha wo resricions, ν 1 = λ 1 and ν = λ, give us marke risk premium in he long erm. Facorizing he common muliplier from all formulas one obains he following raio of equiy and marke risk premia: RP e RP m = ˆν 1 ˆλ 1. This raio is common for all hree uiliy funcions. Furhermore, i does no depend on preference parameers. This raio should be recognized as elasiciy of subsiuion beween dividend and consumpion growh raes. In paricular, when dividends growh raes are idenical o hose of consumpion, his raio is equal o one. If dividends grow bu do no vary across saes, his raio is equal o zero. This goes in conras wih shor-erm premium raio where even for equal dividend growh raes, price dividend raios are sill differen across saes. In shor erm his raio is equal o zero only when fuure asse payoffs, no growh raes, are equal across saes, ν 1 w = ν w

35 1.4 Daa and esimaion Daa descripion The daa for he US annual aggregae real consumpion and he US populaion is aken from Federal Reserve bank of S. Louis 7. Dividing aggregae consumpion by populaion and convering o growh raes I obain real consumpion per person growh rae which plays he role of marke dividend growh series. The daa range spans almos 80 years from 1930 o 009. The daa on sock reurns as well as risk-free and marke reurns were aken from Fama- French daa library 8. I have chosen o work wih decile porfolios sored by book-o-marke BM, dividend yield DP, and size ME which are consruced by Fama and French The reason I resriced my aenion o hese hree sors is ha hese are he only series ha sar a This source conains wo versions of nominal gross reurns: including and excluding dividends. These daa allows o exrac dividend growh raes necessary for empirical invesigaion of his paper. Denoe wo versions of reurns repored as follows: R P i = P i P i 1, R i = P i + D i P i 1 = R P i + D i P i 1. From his I can infer curren dividends: D i = P i 1 R i R P i, and consequenly, dividend growh rae: d i = D i D i 1 1 = P i 1 R i Ri P 1 = R P i R P R i Ri P i 1 Ri 1 P i 1 R i 1 Ri 1 P 1. 7 hp://research.slouisfed.org/fred/ 8 hp://mba.uck.darmouh.edu/pages/faculy/ken.french/daa_library.hml 8

36 These are nominal growh raes and need o be convered o real by subracing inflaion rae. I obain annual inflaion series from he US Bureau of Labor Saisics 9. Call he oal risk free reurn as R f, oal marke reurn as Rm, and ih porfolio reurn as R i. Using hese daa I am able o compue esimae uncondiional expecaion of shor and long excess reurns for each porfolio RP i 1 = log 1 T T R i log =1 1 T T R f, =1 and T RP i = log R i =1 1 T T log R =1 1 T = 1 T T =1 log R i log R f. Marke excess reurn is compued analogously. Descripive saisics for annual consumpion growh, inflaion, marke reurns, and riskfree reurns are given in he Table A.1 on page 107. Means and sandard deviaions of dividend growh raes and annual reurns on hiry sored porfolios are given in he Table A. on page Esimaion In order o quanify implicaions of he model I have chosen a differen approach ha was used by Mehra and Presco 1985 who used he one-o-one connecion beween model parameers and growh rae momens o mach hem wih hisorical US consumpion daa. There are hree parameers o mach, µ, δ, p, and hree empirical momens, mean, sandard deviaion, and firs order auocorrelaion. In his paper I use insead he approach suggesed by Tauchen 1986 o esimae Markov ransiion marix and growh raes of consumpion and dividends. Two feaures of heir esimaion procedure are decisive facors in my choice. Firs, he mehod works for any finie number of saes, no jus wo. Second, he mehod works boh for univariae ime series and 9 hp:// 9

37 mulivariae which allows o esimae join dynamics of consumpion and dividends. The essence of he mehod is o fi VAR1 o he daa and esimae model errors ha are assumed o be independen. These errors are used o esimae is univariae densiies for example by non-parameric kernel mehods. I used nonparameric densiy esimaor wih defaul bandwidh and Gaussian kernel provided by MaLab. Equally spaced grid is hen used o assign probabiliies o he finie number of inervals on he grid. Transiion probabiliies for he original processes are backed ou from he VAR srucure and is parameer esimaes. Available daa on dividend growh raes on monhly ime scale is exremely noisy in comparison o consumpion growh daa. So I work wih he annual daa only. 30

38 1.5 Resuls The analysis of model soluion is performed hrough he sensiiviy analysis of asse reurn momens. Table A.3 on page 108 shows he benchmark parameer values. Given hese values I vary each parameer over he large inervals in order o undersand he effecs of each one on asse pricing implicaions. This secion is organized as follows. Secion Secion analyzes numerical decomposiion of he SDF marix. Secion Secion 1.5. breaks down marke reurn ino permanen and ransiory componens and shows heir dependence on model parameers. Secion Secion looks a he erm srucure of risk premia and is componens Transiion marix decomposiion In his secion I discuss numerical resuls of he SDF marix decomposiion. I will show ha he permanen componen, or subjecive probabiliy ransiion marix, is almos he same for expeced and EZ uiliy funcions. On he oher hand, he subjecive probabiliy disribuion implied by GDA is iled oward bad economy oucomes. Table A.5 on page 109 repors SDF marices and is decomposiion for each of he hree uiliy funcions. The firs row and firs column of each marix corresponds o he low consumpion growh. Firs of all noe ha he modificaion marix of EZ uiliy, K, has he diagonal elemens very close o one and off diagonal elemens being almos reciprocal of each oher. This numerical resul simply follows from consrucion of his marix as a raio of marke price-dividend raios. As i was shown in Secion 1.3 his modificaion of he SDF is represenable in he form S Z = W θ S X W + I θ, where W is he diagonal marix wih marke price-dividend raios, and θ = 1 aσ σ 1. In he same secion I have argued ha as long as P/D raios are large and no volaile, he modificaion 31